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Baxter operators in Ruijsenaars hyperbolic system IV. Coupling constant reflection symmetry

N. Belousov, S. Derkachov, S. Kharchev, S. Khoroshkin

Abstract

We introduce and study a new family of commuting Baxter operators in the Ruijsenaars hyperbolic system, different from that considered by us earlier. Using a degeneration of Rains integral identity we verify the commutativity between the two families of Baxter operators and explore this fact for the proof of the coupling constant symmetry of the wave function. We also establish a connection between new Baxter operators and Noumi-Sano difference operators.

Baxter operators in Ruijsenaars hyperbolic system IV. Coupling constant reflection symmetry

Abstract

We introduce and study a new family of commuting Baxter operators in the Ruijsenaars hyperbolic system, different from that considered by us earlier. Using a degeneration of Rains integral identity we verify the commutativity between the two families of Baxter operators and explore this fact for the proof of the coupling constant symmetry of the wave function. We also establish a connection between new Baxter operators and Noumi-Sano difference operators.
Paper Structure (20 sections, 22 theorems, 350 equations)

This paper contains 20 sections, 22 theorems, 350 equations.

Table of Contents

  1. Introduction
  2. Ruijsenaars system and $Q$-operator
  3. Reflection $g \rightarrow g^*$ and $Q^*$-operator
  4. Wave function and local relations
  5. Relation to Noumi-Sano operators
  6. Local relations
  7. $Q^*Q$ commutativity
  8. $Q^*\Lambda$ exchange relation
  9. $\Lambda^*\Lambda$ exchange relation
  10. Eigenfunctions
  11. Eigenfunctions of $Q^*$-operator
  12. Wave function $g \rightarrow g^*$ symmetry
  13. Noumi-Sano difference operators
  14. The double sine function
  15. A degeneration of Rains integral identity
  16. ...and 5 more sections

Key Result

Theorem 1

Under conditions I0a, I0b, nu* the two families of Baxter $Q$-operators commute The kernels of the operators in both sides are analytic functions of $\lambda, \rho$ in the strip

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 5
  • Theorem 5
  • ...and 23 more